On character table of Clifford groups

TL;DR

This work delivers explicit character tables for the Clifford groups $\mathcal{C}_n$ at $n=1,2,3$ by leveraging a concrete presentation and GAP computations, enabling direct decomposition of tensor powers of the matrix representation. It establishes a robust framework: a full presentation of $\mathcal{C}_n$, a precise determination of normal subgroups, and the resulting structural results for $n\geq 3$, including that $\mathcal{C}_n=[\mathcal{C}_n,\mathcal{C}_n]$ and $\mathcal{P}_n$ is the unique nontrivial normal subgroup, with $\mathcal{C}_n/\mathcal{P}_n\cong Sp(2n,2)$ being simple. The paper also provides a byproduct presentation of $Sp(2n,2)$ and demonstrates the faithfulness of the $\mathcal{M}_{2^n}$ representation for $n\geq 3$, together with concrete tensor-product decompositions that recapitulate known results and reveal new phenomena for higher tensor powers. By integrating theoretical group presentations with computational tools, the work advances practical access to Clifford group representations, with implications for quantum information tasks such as unitary designs and error correction. Ma's related Clifford theory is mentioned as a path to extending the character tables to larger $n$ via lifting from $\mathcal{C}_{n-1}$ and $Sp(2n,2)$.

Abstract

Based on a presentation of $\mathcal{C}_n$ and the help of [GAP], we construct the character table of the Clifford group $\mathcal{C}_n$ for $n=1,2,3$. As an application, we can efficiently decompose the (higher power of) tensor product of the matrix representation in those cases. Our results recover some known results in [HWW, WF] and reveal some new phenomena. We prove that when $n \geq 3$, (1) the trivial character is the only linear character for $\mathcal{C}_n$ and hence $\mathcal{C}_n$ equals to its commutator subgroup, (2) the $n$-qubit Pauli group $\mathcal{P}_n$ is the only proper non-trivial normal subgroup of $\mathcal{C}_n$, (3) the matrix representation $\mathcal{M}_{2^n}$ is a faithful representation for $\mathcal{C}_n$. As a byproduct, we give a presentation of the finite symplectic group $Sp(2n,2)$ in terms of generators and relations.

Theorems & Definitions (30)

• Definition 2.1
• Definition 2.2
• Lemma 3.1
• Corollary 3.2
• proof
• Remark 3.3
• Theorem 4.1
• Remark 4.2
• Remark 4.3
• proof